Convergent Series

I have been thinking about this problem:
Determine whether the following series are convergent in [tex]\left(C[0,1],||\cdot ||_{\infty}\right)[/tex] and [tex]\left(C[0,1],||\cdot ||_{1}\right)[/tex].
when
[tex]f_n(t)=\frac{t^n}{n}[/tex]

In the supremum norm, this seems pretty straightforward, but in the integral norm I am confused since,
[tex]\left\|\sum\frac{t^n}{n}\right\|_1\leq\sum\left\|\frac{t^n}{n}\right\|_1=\sum\int_0^1\frac{t^n}{n}dt=\sum\left[\frac{t^{n+1}}{n^2+n}\right]_0^1=\sum\frac{1}{n^2+n}<\sum\frac{1}{n^2} [/tex]
and, I think this converges as [tex]n\rightarrow\infty[/tex], but our instructor said this did not converge, or maybe I heard him incorrectly. So, does this converge? He asked us to show the series is Cauchy and that the limit is not in the space as well. What am I missing?

But, because the space [tex]\left(C[0,1],\|\cdot\|_1\right)[/tex] is not complete all we know is that [tex]f_n(t)[/tex] is Cauchy. I was missing the part about the space being complete.
Also, you can see that limit is not an element of C[0,1].